Is it permissible for me to make up for my late father`s missed fasts, and should I make an intention to this end by saying: “I intend to make up for my late father`s missed fasts .”

It is permissible to fast on behalf of the deceased father in order to make up for his missed fasts, and you should make the intention for offering fast from night time, but uttering the intention isn`t a condition.

Is a woman`s neck an Awrah(private part)?

It is an Awrah, and it is forbidden for her to reveal it before non-Mahrams(Marriageable men).

What is the ruling on singing and listening to it while accompanied and unaccompanied by musical instruments?

Singing and listening to it unaccompanied by musical instruments is permissible as long as it isn`t practiced as a profession and is done by a man to other men, or a woman to other women with maintaining that no temptations are involved. The words of the song should be Islamic, and if temptation was likely to happen, then the singing becomes definitely forbidden. Temptation here means: whatever leads to committing a sin. In Islamic Sharia, singing accompanied by musical instruments such as the lute and the like is forbidden, and listening to it as well. The reason behind that is the fact that the musical instrument itself is forbidden. However, striking the tambourine and singing Islamic songs are permissible in weddings, circumcisions and the like.

An engineer has to work in a circuit system of a non-Islamic bank. What is the ruling on the income she receives from doing this job?